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Simplifying 12x2 + 31x + -20 = 0 Reorder the terms: -20 + 31x + 12x2 = 0 Solving -20 + 31x + 12x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -1.666666667 + 2.583333333x + x2 = 0 Move the constant term to the right: Add '1.666666667' to each side of the equation. -1.666666667 + 2.583333333x + 1.666666667 + x2 = 0 + 1.666666667 Reorder the terms: -1.666666667 + 1.666666667 + 2.583333333x + x2 = 0 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + 2.583333333x + x2 = 0 + 1.666666667 2.583333333x + x2 = 0 + 1.666666667 Combine like terms: 0 + 1.666666667 = 1.666666667 2.583333333x + x2 = 1.666666667 The x term is 2.583333333x. Take half its coefficient (1.291666667). Square it (1.668402779) and add it to both sides. Add '1.668402779' to each side of the equation. 2.583333333x + 1.668402779 + x2 = 1.666666667 + 1.668402779 Reorder the terms: 1.668402779 + 2.583333333x + x2 = 1.666666667 + 1.668402779 Combine like terms: 1.666666667 + 1.668402779 = 3.335069446 1.668402779 + 2.583333333x + x2 = 3.335069446 Factor a perfect square on the left side: (x + 1.291666667)(x + 1.291666667) = 3.335069446 Calculate the square root of the right side: 1.82621725 Break this problem into two subproblems by setting (x + 1.291666667) equal to 1.82621725 and -1.82621725.Subproblem 1
x + 1.291666667 = 1.82621725 Simplifying x + 1.291666667 = 1.82621725 Reorder the terms: 1.291666667 + x = 1.82621725 Solving 1.291666667 + x = 1.82621725 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.291666667' to each side of the equation. 1.291666667 + -1.291666667 + x = 1.82621725 + -1.291666667 Combine like terms: 1.291666667 + -1.291666667 = 0.000000000 0.000000000 + x = 1.82621725 + -1.291666667 x = 1.82621725 + -1.291666667 Combine like terms: 1.82621725 + -1.291666667 = 0.534550583 x = 0.534550583 Simplifying x = 0.534550583Subproblem 2
x + 1.291666667 = -1.82621725 Simplifying x + 1.291666667 = -1.82621725 Reorder the terms: 1.291666667 + x = -1.82621725 Solving 1.291666667 + x = -1.82621725 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.291666667' to each side of the equation. 1.291666667 + -1.291666667 + x = -1.82621725 + -1.291666667 Combine like terms: 1.291666667 + -1.291666667 = 0.000000000 0.000000000 + x = -1.82621725 + -1.291666667 x = -1.82621725 + -1.291666667 Combine like terms: -1.82621725 + -1.291666667 = -3.117883917 x = -3.117883917 Simplifying x = -3.117883917Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.534550583, -3.117883917}
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